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# Population Dynamics - Part 4

### The Logistic Map

### Finding Equilibrium Points

### Different Cases of Stability

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The logistic map is the discrete case of the logistic equation, given by: $\frac {\mathrm{d}y}{\mathrm{d}t}=ry(1-\frac{y}{Y})$

We then approximate to deduce the discrete case:$$ \begin{align*} \frac{y_{n+1}-y_n}{\Delta t} &\approx ry_n\left(1-\frac{y_n}{Y}\right) \\ y_{n+1} &\approx r \Delta t y_n \left(1-\frac{y_n}{Y}\right)+y_n \\ y_{n+1}&=(1+r \Delta t)y_n-\frac {r\Delta t}{Y}{(y_n)}^2 \\ y_{n+1}&=(1+r \Delta t)y_n\Bigg( 1-\bigg(\frac{r\Delta t}{1+r\Delta t}\bigg)\frac{y_n}{Y}\Bigg) \end{align*} $$

Let $\lambda=1+r\Delta t$ and $x_n=\frac {r\Delta t}{1+r \Delta t} \frac {y_n}{Y}$ . Then our equation becomes: $$x_{n+1}=\lambda x_n (1-x_n) $$ This is the logistic map. We can also think of it as a function $x_{n+1}=f(x_n)$.

**Question:** A fixed point implies $x_{n+1}=x_n$ . Find the fixed points by solving $$ \lambda x_n (1-x_n) = x_n $$ To determine the stability of these points, we are going to find the stability, by investigating the function for values nearby the equilibrium points.

Start by supposing that $x_n=X$ is a fixed point. This means that $f(X)=X$.

To find a value near the equilibrium point, let $x_n=X+\epsilon{_n}$ where $\epsilon_n < < 1$. Then using the Taylor expansion: $$ \begin{align*} x_{n+1}&=f(x_n) \\ X+\epsilon_{n+1} &= f(X+\epsilon_n) \\ &=f(X)+\epsilon_n f'(X)+... \end{align*}$$

We neglect the higher-order terms to get: $$X+\epsilon_{n+1}=f(X)+\epsilon_n f'(X)$$ Now from above we saw that $f(X)=X$ , so we can simplify to get: $$\epsilon_{n+1} \approx f'(X) \epsilon_n$$ A fixed point, *X*, is then stable if: $\Bigg|\frac{\epsilon_{n+1}}{\epsilon_n}\Bigg | =\Bigg |f'(X)\Bigg | < 1$

**Question:** Given that $f'(x)=\lambda-2\lambda x$ , find the stability of the fixed points $x_n=0$ and $x_n=1-\frac{1}{\lambda}$

Below are some graphs of the logistic map for different values of $\lambda$ .

**Case 1: $\lambda< 1$**

Only fixed point is 0, which is stable:

**Case 2: $1< \lambda < 2$**

Unstable fixed point at 0 and stable fixed point at $1-\frac{1}{\lambda}$

**Question**: Can you find the stability for the case $2< \lambda < 3$ ?

Below is a picture of some fantastic fractal behaviour which occurs for $3< \lambda< 4$.

**Question:** Can you relate these values of $\lambda$ to what would actually be occuring in a population of organisms?

bioNRICH is the area of the stemNRICH site devoted to the mathematics underlying the study of the biological sciences, designed to help develop the mathematics required to get the most from your study of biology at A-level and university.

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